Question: Find the point of tangency of the parabolas $y = x^2 + 15x + 32$ and $x = y^2 + 49y + 593.$
Solution: Adding the equations, we get
\[x + y = x^2 + 15x + 32 + y^2 + 49y + 593,\]or $x^2 + 14x + y^2 + 48y + 625.$  Completing the square in $x$ and $y,$ we get
\[(x + 7)^2 + (y + 24)^2 = 0.\]We can check that $\boxed{(-7,-24)}$ lies on both parabolas, so this is the point of tangency.